Horava-Witten String Theory
The Horava-Witten String Theory (also known as the Type HW String Theory), is a strongly-coupled 10-dimensional Superstring Theory. Being a theory with strong-coupling, it cannot be accurately studied perturbatively, as the peturbation would diverge and not be renormalisable. As with all strongly coupled string theories, it is a theory of "heavy" (i.e. strongly coupled) D1-Branes, as opposed to a theory of "light" (i.e. weakly coupled) F1 strings. The theory is named after the physicists Petr Hořava and Edward Witten. . Relation with other string theories The Hořava–Witten String Theory is S-Dual to the Type HE String Theory, by definition. Since the Type HE String Theory is T-Dual to the Type HO String Theory, the Hořava–Witten String Theory is also U-dual to the Type HO String Theory. Since the Type HO String Theory is S-Duality to the Type I String Theory, the Hořava–Witten String Theory is also S-Dual to the T-dual of the S-dual of Type I String Theory; i.e. it is T-Dual to the Type I String Theory. It is commonly stated that the T-Dual of the Type I String Theory is the 9-dimensional "Type I' String Theory". However, rather than being a T-Duality, this is more of an equivalence, because Type I' String Theory is simply the Hořava–Witten String Theory compactified on a circle of zero radius. By the definition of T-Duality, uncompactified Type I String Theory, i.e. Type I String Theory compactified around a circle of infinite radius, is equivalent to its T-dual (Hořava–Witten String Theory) compactified around a circle of 0 radius, which is the 9-dimensional Type I' String Theory. This is actually an instance of the Holographic principle, an equivalence between a D dimensional theory, and a D-1 dimensional theory. Hořava–Witten String Theory played a crucial role in the discovery of the underlying M-Theory, for the Hořava–Witten String Theory is M-Theory compactified on a line segment, i.e., with a Hořava–Witten Boundary. Action principle As the Hořava–Witten String Theory is a strongly-coupled String Theory, ordinary actions like the Polyakov and RNS fail, as they are perturbative. Instead, one needs to use the AdS/CFT correspondence, an instance of the Holographic principle. Through AdS/CFT, Hořava–Witten String Theory is described by a Matrix String Theory. The statement defining the action principle is as follows: :: Hořava–Witten String Theory in Anti-de Sitter space is exactly described by the non-relativistic Quantum Super-Yang-Mills Theory with gauge group O(N) , with the N eigenvalues of the operators, (witheigenvector, being the state vector) interpreted as points on the string; the fundamental "string-bits".. Thus, the Lagrangian Density is given by: \mathsf{\mathcal{L}}=\frac{1} }\left( \frac{1}{2}\operatorname{tr}\left( \frac{\partial }{\partial \sigma }\frac{\partial }{\partial \tau } \right)-\frac{1}{4}\operatorname{tr}\left( , \right]}^{2}} \right)+ \left[ , \right] \right)\mbox{ (1)} Where the bosonic matrices X and their fermionic superpartners {\mathbf{\psi}} are O(N) generators. This Lagrangian density immediately allows for a non-peturbative formulation of Type HE String Theory. By S-Duality, swapping \frac1{g_s} with g_s yields the Type HE String Theory. \mathsf{\mathcal{L}}=g_s\left( \frac{1}{2}\operatorname{tr}\left( \frac{\partial }{\partial \sigma }\frac{\partial }{\partial \tau } \right)-\frac{1}{4}\operatorname{tr}\left( , \right]}^{2}} \right)+ \left[ , \right] \right)\mbox{ (2)} Replacing O(N) with U(N) for Equation (1) results in the S-dual of the Type IIA String Theory. \mathsf{\mathcal{L}}=\frac{1} }\left( \frac{1}{2}\operatorname{tr}\left( \frac{\partial }{\partial \sigma }\frac{\partial }{\partial \tau } \right)-\frac{1}{4}\operatorname{tr}\left( , \right]}^{2}} \right)+ \left[ , \right] \right)\mbox{ (3)} And then again, taking the S-Dual results in the Type IIA String Theory: \mathsf{\mathcal{L}}=g_s\left( \frac{1}{2}\operatorname{tr}\left( \frac{\partial }{\partial \sigma }\frac{\partial }{\partial \tau } \right)-\frac{1}{4}\operatorname{tr}\left( , \right]}^{2}} \right)+ \left[ , \right] \right)\mbox{ (4)} Notice that the Lagrangian densities look the same, but they are computationally different, as the matrices are taken to be in U(N) instead. Also, while the Lagrangian densities for the S-Dual of the Type IIA String Theory and the Type IIA String Theory look different, they result in the same theory, only with the different fundamental object; D1 branes and F1 Strings respectively. Category:String Theory